Verify Gauss divergence theorem of F=xi+yj+zk over the sphere x^2+y^2+z^2=a^2

zaviknuogg 2022-09-22 Answered
Verify Gauss divergence theorem of F = x i + y j + z k over the sphere x 2 + y 2 + z 2 = a 2
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Answers (1)

Ashlynn Delacruz
Answered 2022-09-23 Author has 9 answers
Let S a be your sphere, and B a the enclosed ball. For each point r S a the outwards unit normal n is given by n = r a . Furthermore F ( r ) = r . Since r r = a 2 on S a it follows that
S a F n d ω = S a r r a d ω = a S a d ω = 4 π a 3   .
On the other hand, d i v ( F ) 3, and therefore
B a d i v ( F ) d v o l = 3 v o l ( B a ) = 4 π a 3   .
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